The time of oscillation of a bar pendulum is shown to be practically unaffected by the use of a carriage supporting the knife-edge and sliding along the bar; the accuracy of the usual laboratory exercise on the bar pendulum is considerably increased. With a stop-watch showing tenths of a second the value of g can be determined quickly to within a few parts in 10,000.
If a flat, rigid body is pivoted about any point other than its center of mass and displaced by a small angle, the body will execute SHM (Simple Harmonic Motion).
Thederivation is similar to that of a simple pendulum since one can consider all the mass M to be located at the body's center of mass. Thena physical pendulum looks like a simple pendulum except that its moment of inertia is found using the parallel axis theorem.
Icm =the body's moment of inertia about its center of mass.
h = Distance from pivot point to the center of mass.
M = Mass of the body.
Ifthe pivot joint is frictionless then the net torque acting on the planar object is given by the force of gravity perpendicular to lever arm, Mg sin(q), times the length of the lever arm, h:
Whenthe angle of oscillation is small then the value of sin(q) and q or nearly the same provided q is measured in radians. Using this approximation, the above torque equation can be solved for the angular acceleration,
Since a =d2q/dt2, this equation is structurally similar to the differential equation for any type of SHM,
Matching terms with SHM equations,
Here w' is instantaneous angular velocity of the body while w is angular frequency of the body's SHM. They are not the same. Moreover, w is constant while w' varies as the body oscillates back and forth.